On the Kummer congruences and the stable homotopy of $B$U

Abstract:

We study the torsion-free part of the stable homotopy groups of the space $BU$, by considering upper and lower bounds. The upper bound is furnished by the ring $P{K_{\ast }}(BU)$ of coaction primitives into which $\pi _{\ast }^S(BU)$ is mapped by the complex $K$-theoretic Hurewicz homomorphism \[ \pi _{\ast }^S(BU) \to P{K_{\ast }}(BU).\] We characterize $P{K_{\ast }}(BU)$ in terms of symmetric numerical polynomials and describe systematic families of elements by utilizing the classical Kummer congruences among the Bernoulli numbers. For a lower bound we choose the ring of those framed bordism classes which may be represented by singular hypersurfaces in $BU$. From among these we define families of classes constructed from regular neighborhoods of embeddings of iterated Thom complexes in Euclidean space. Employing techniques of duality theory, we deduce that these two families correspond, except possibly in the lowest dimensions, under the Hurewicz homomorphism, which thus provides a link between the algebra and the geometry. In the course of this work we greatly extend certain $e$-invariant calculations of J. F. Adams.